Questions tagged [quantum-chemistry]
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Thermodynamics and Gibbs energy
The equation for the Gibbs free energy is: $\Delta G = \Delta H - T\Delta S.$
$\Delta H$ is the amount of energy difference between the initial state (characterized by $U_{i}$, $S_{i}$, and $V_{i}$) ...
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How is energy conserved in formation of chemical bonds?
I understand that when an H$_2$O molecule is formed, it has lesser total energy than the constituent (2H and O) atoms. I also understand that the potential energy (PE) will be reduced when the atoms ...
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Measurement of dipole electric field generated by a polar molecule
I'm trying to understand the nature of polar molecules. Have there been experiments where we can say to have actually measured the dipole electric field generated by a polar molecule such as hydrogen ...
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Is Hartree-Fock energy minimisation an iterative or a Lagrange undetermined process?
Suppose the following is about a quantum system of N electrons.
In the canonical Hartree-Fock method, the goal is to find single - electron orbitals $\phi_{1}, ...,\phi_{N}$ such that $\langle \psi_{...
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Number of summations for a Hartree-Fock (single-orbital as basis functions) method
The Hamiltonian for a quantum $N$ - body system:
$$\hat{H} = - \frac{1}{2}\sum_{i}^{N}\nabla_{i}^{2}-\sum_{m;i=1}^{N}\frac{z_{m}}{r_{im}}+ \sum_{i;j>i}\frac{1}{r_{ij}}.$$
The first two terms can be ...
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Can a matter enter do work without being consumed up? [closed]
I had a tricky question in mind. So, I wanted to use the example of Catalysts (In chemical reactions) as an example. It's said that when catalysts enter a chemical reaction, they're Never consumed up. ...
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Where is the kinetic energy functional error in the exchange-correlation functional of DFT?
The Kohn-Sham DFT energy functional is:
$$E[\rho]=T^S[{\varphi_i}]+J[\rho]+M[\rho]+E^{xc}[\rho]$$
with the kinetic energy functional of non-interacting electrons $T^S$, the Hartree functional $J$, the ...
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Why using real wave functions instead of complex ones?
I have already seen similar questions asked in the site (like this or this), but I don't feel that my question has been fully addressed.
I understand that orbitals $np_x$ and $np_y$ are linear ...
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On completeness of a set of commuting operators for homonuclear diatomic molecules
The electronic Hamiltonian for a homonuclear diatomic molecule is $$\hat{H}=-\sum_{i=1}^N \frac{\hbar^2}{2m} \nabla^2_i -\sum_{i=1}^N \frac{Z_Ae^2}{4\pi\epsilon_0|\vec{r}_i-\vec{R}_A|} -\sum_{i=1}^N \...
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Why does the electron distribution around an atom change over time?
I am reading about Van der Waals forces and what I understood is that the electron distribution around an atom changes over time causing transient dipoles to form that induce transient dipoles in ...
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Density functional theory: logical steps from Hohenberg-Kohn theorems to Kohn-Sham equations
I'm trying to learn density functional theory (DFT), using Engel & Dreisler, Sholl & Steckel, and Wikipedia as 3 different sources of information. Although I have made some progress, I am ...
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Deducing physical properties of substances with physics
Inspired by this question (Why does amount of protons define how matter is?) I’m wondering how one can deduce the physical properties of an element with physics. For example, it is known that sodium ...
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Gibbs energy calculations for compounds
I am studying machine learning during my master's degree and have a simple task. I am using the FactSage Pure Substance Database (https://www.crct.polymtl.ca), which provides a list of phases for a ...
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Is it possible to calculate the average kinetic energy of an atomic orbital in an heavy atom? [closed]
I am trying to calculate cross sections for electron impact ionization in heavy atoms (Xe and I) using a BEB technique. To do this I need, for each occupied orbital, occupancy number, binding energy ...
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Expressing overlap between two spinors as an integral and a sum
I was trying to go through some material involving the Hartree–Fock method for tackling the many electron problem. In most resources, namely in Lucjan Piela (pg 334) that we can the overlap between ...
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What is the physical origin of two Hydrogen atoms repelling at short distance?
Consider two isolated $\rm H$ atoms. They can be in different spin states, but the different states are degenerate. Now move these $\rm H$ atoms together. Now the singlet state will become ...
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Where does this (hydrogen molecule energy) graph come from?
I was thinking about the good old question of 'Why do molecules have lower energy than the atoms?'
And in a video (around 6:15), this good old energy graph is shown, which is stated as the 'answer' to ...
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Dirac equation for solving $\rm H^-$ anion
Let's try to solve 1 dimensional Dirac equation for a $\rm H^-$ anion. If you solve the time independent Dirac equation and allow motion of electrons only in the $x$ axis you get:
$$-\chi\frac{d\Psi_{...
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How do nuclear physicists know for certain if a very short-lived nucleus has been formed (a new isotope)?
Some very short-lived nuclides (sometimes called 'resonances') are nevertheless considered to have actually existed, if only very very briefly....
How do nuclear physicists know the isotope actually ...
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Why does Fock operator in Hartree-Fock method have the molecule symmetry?
I am reading Szabo - 1967 - Modern Quantum Chemistry. In Chapter 3 P122, it mentions that the spin orbitals $\left|\chi_a\right\rangle$ obtained by the Fock operator
$f\left|\chi_a\right\rangle=\...
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How is the band gap really defined and calculated?
When I look into literature I am quite confused on how the band gap of a semiconductor is defined. One statement I often read goes something like that:
The band gap of a semiconductor is the energy ...
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How do I interpret the single electron Hamiltonian contribution in the Roothan-Hall equation?
I am currently going through Molecular Modelling: Principles and Applications by Andrew R. Leach (2nd Edition).
The Hamiltonian operator for an electron is given in equation (2.125) as
$$\mathscr{H}^{...
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Quantum simulation of molecular spectroscopy: why?
Molecular spectroscopy, e.g. through light-matter interaction, is a powerful tool to measure molecular properties such as bond lengths, energy levels, vibration frequencies, ... which give insights ...
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Can you burn a quantum fluid?
Perhaps more generally, can you have a chemical reaction between a quantum fluid (a sodium Bose-Einstein condensate for example) and another quantum fluid or a non-quantum fluid?
Or as a silly ...
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Is Hartree-Fock (HF) a ground state theory?
To calculate the orbitals in Hartree-Fock (HF) theory we imply the variations principle, so we try to find the wavefunction which minimises the energy $\left<\psi|H|\psi\right>$.
This ...
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Why does Hartree-Fock (HF) theory even work?
Let’s say we have $N$ electrons and we want to derive the Hartree-Fock (HF) equations. The first step would be to define a Slater determinant of $N$ electrons:
$$\psi(x_1,x_2,… x_N) = \frac{1}{\sqrt{...
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From molecular orbitals to band diagram
Let’s say we have a periodic crystal structure. We could, in theory, treat this system as if it were a large molecule. Therefore, we could use Hartree Fock theory or other methods to get the molecular ...
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Why is there no exchange interaction when there is no coulomb interaction?
The Hartree-Fock energy may be written as
$$
E_{HF}= \langle\Psi| H |\Psi\rangle = \sum_{a} \langle a| h |a \rangle + \frac{1}{2}\sum_{ab} \big( [aa |bb] - [ab|ba] \big),$$
with the exchange term
$$[...
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Classical optical activity parallel to the helical axis of a molecule/spring
The image below is from Kauzmann's Quantum Chemistry.
It is an intuitive classical explanation of optical rotation due to light hitting a helical structure (usually a molecule) perpendicular to the ...
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General form of a fermionic Hamiltonian in second quantization
I am a new to quantum chemistry. I've been reading this paper and it seems like the equation (1) in the paper which is,
$H = \sum_{pq} T_{pq} a_p^\dagger a_q + \sum_p U_pn_p + \sum_{pq} V_{pq} n_p n_q$...
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Ammonia electric dipole moment according to Philip Warren Anderson
In his paper "More is Different", Philip Warren Anderson claims to be in posession of a proof by which any stationary state of an Ammonia molecule (or anything really, again according to him,...
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What is the reason for Avagadro's Law?
From outside the quantum stuff I'm tiredelessly learning, I've been reminded about Avagadro's Law
— i.e., the fact that any molecular gas has the same count of molecules in the same volume and ...
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Potential Energy of the Helium Atom
So I just learned how to solve the Schroedinger Equation for the Hydrogen atom. The set up looked like this: $$\left[\dfrac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\dfrac{\partial}{\partial r}\...
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How did the Aufbau principle come to be? [duplicate]
How did the Aufbau principle come to be? Being that my education on chemistry doesn't go past highschool, i was never sure how the following diagram was conceived:
I would like to understand how the ...
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Is it justified to take orbital as just a "dispersed region of charge" for applications in Organic chemistry?
When I read organic chemistry book in the past, it seems so a heuristic used to explain and understand reaction mechanism is that negative charge attracts positive charge (or indirectly as ...
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Separatibility of wave-function in Schrodinger equation describing motion of particle in spherical shell [closed]
Question.
Why wave-function describing rotation of particle around spherical surface is separable?
Why solution obtained assuming function is separable is general solution?
I am interested in proof of ...
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Are there some references to periodic Hartree-Fock calculation?
I want to learn Hartree-Fock(HF) calculation for periodic system, but didn't find some reference. And also have some confusion about it now.
In one book(Quantum Chemistry of Solids: The LCAO First ...
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Electrostatic self-interaction of two multivariate Gaussian charge densities
I'm trying to figure out the following question:
If $\mathcal{N_1}(\vec x|\vec \mu_1, \Sigma_1)$ and $\mathcal{N_2}(\vec x|\vec \mu_2, \Sigma_2)$ are two arbitrary multivariate Gaussian charge ...
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potential energy of two atom Molecule in case of ionic bond
The formula for the potential energy of a two atomic Molecule in case of a ionic bond is given by:
$$E_p(r)= -\frac{e^2}{4 \pi \epsilon_0 r} + \frac{b}{r^9}$$
What is the physical meaning of these ...
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What eigenstates of the Fock operator gives the Hartree Fock ground state?
In the Hartree Fock method with $N$ electrons, the Fock operator is a one-electron operator defined by
$$f := h + \sum_{n=1}^N J_n - K_n,$$
where $h = -\frac{1}{2} \nabla^2 + V_{el-nu}$ is the one-...
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Multi-electron Bloch bundles over powers of the Brillouin torus?
The quantum states of single electrons in a crystal famously form a (Hilbert) vector bundle over the Brillouin torus ${\mathbb{T}}^d$ -- the Bloch bundle.
Does this theory usefully generalize to ...
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Finding a bound on ground state energy derivatives of molecule
Part of a project I'm working on involves exploring the smoothness of the ground state energy $E(x)$ of a molecule where $E$ is parameterized by the geometry of the molecule. In particular, my goal ...
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What is the single active electron approximation?
I keep stumbling upon this approximation while reading papers on atomic physics, but I couldn't find a proper description of what it is in any textbook or pedagogical paper. The closest I could find ...
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QM: How do electrons affect each other's potential energy?
I'm trying to learn quantum mechanics. And I'm confused.
The question is, do electrons affect each other's wave functions, i.e. potential energies?
How do they affect each other orbital shapes, how do ...
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The uncertainty principle and the quantization of electron momentum in orbitals - Why are larger energy gaps considered more uncertain?
I am a chemist trying to understand how the uncertainty principle can be applied to the mechanics of atomic and molecular orbitals in order to have better intuitions about stability and reactivity. ...
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Is ionization energy non-negotiable?
$$\text{2Na(s)}+\text{Cl}_2\text{(g)}\longrightarrow \text{2NaCl(s)}\tag{1}$$
This is a spontaneous reaction ($\Delta G<0$). 2 Na atoms donate 2 electrons to 2 Cl atoms.
However, Na still has a ...
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How does the concept of activation energy work for interatomic collisions?
Say we have a Lennard-Jones -esque potential for colliding atoms. Say this potential admits a bound state at a lower energy than the kinetic energy of the colliding atoms. So forming this molecule ...
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Does the copper normalize?
I am reading Concepts of Elemental Particle Physics. In it, it describes what happens when a high energy muon is shot into a copper rod. An electron is is knocked out of its orbit and and photons ...
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Excitation of electrons from $3p$ to $3d$ in sulphur
In my book it is shown that sulphur in its excited state forms 6 covalent bonds with Fluorine to form SF6 but after the electron is excited does it not dexcite from $3d$ back to $3p$ and $3s$ by ...
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Why do some atoms like Mg have several ionization energy levels?
In Dopant Energy Levels in Crystalline Silicon (which asks for a password, but is partially readable if you click "cancel") appears the following image and caption:
Figure 2777b shows the ...
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